THE NONLINEAR EVOLUTION OF INVISCID GORTLER VORTICES IN 3-DIMENSIONALBOUNDARY-LAYERS

The nonlinear development of inviscid Gortler vortices in a three-dime nsional boundary-layer flow is considered by extending the theory of B lackaby et al., who consider the closely related problem concerning th e nonlinear development of disturbances in stratified shear flows. The inviscid Gortler modes considered are initially unstable (and hence g rowing) based on linear theory; however, following others, we assume t hat the effects of boundary-layer spreading result in them evolving in a linear fashion until they reach a stage where their amplitudes are sufficiently large, and their growth rates have diminished sufficientl y, such that their subsequent evolution can be considered within the f ramework of a weakly nonlinear theory based on a nonlinear, non-equili brium critical-layer theory. As with the closely related stratified-sh ear flow problem, three possible nonlinear integro-differential evolut ion equations for the disturbance amplitude should arise: however, it is found that only two of these are in fact possible. One of the possi ble integro-differential evolution equations has a cubic-nonlinearity due to supercriticality (non-neutrality) effects, while the other ampl itude evolution equation has a quintic nonlinearity but is only releva nt for larger sizes of disturbance. Thus, in this paper, attention is concentrated on the former, since this equation is appropriate earlier in the evolution process of the Gortler modes, Numerical results are presented which demonstrate that variations in the level of cross-flow present in the underlying flow have a significant effect on the nonli near problem, as the!: do on the linear problem. It is: found that the consideration of a spatial evolution problem (as opposed to a tempora l stability approach adopted in the above paper) leads to significant changes to the resulting evolution equations.